Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel

TL;DR

This paper studies the long-term behavior of solutions to a class of nonlocal Fokker-Planck equations with heavy-tailed kernels, demonstrating uniform convergence to equilibrium regardless of the scaling parameter and fractional index.

Contribution

It introduces a novel analysis combining generalized central limit theorems and Harris's theorem to establish uniform convergence results for these equations.

Findings

Exponential convergence to equilibrium with rate independent of parameters

Uniform-in-time convergence as the scaling parameter approaches zero

Recovery of limiting equations as the fractional index approaches one

Abstract

We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker--Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter $\e\in(0,1]$ and a fractional index $s\in(1/2,1)$. By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris's theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both $\e$ and $s$. This allows us to show uniform--in--time convergence for both $\e\to 0$ and $s\to1$ recovering the limiting equations.